Using the Klein–Gordon equation (2), and expressions (3) for the stress-energy tensor and the corresponding operator, we can write
Substituting Equation (31) into the -dimensional version of the Einstein–Langevin Equation (14), taking into account that satisfies the semiclassical Einstein equation (7), and substituting expression (32), we can write the Einstein–Langevin equation in dimensional regularization as
and the fact that the first term on the right-hand side of this identity is real, whereas the second one is pure imaginary. Once we perform the renormalization procedure in Equation (34), setting will yield the physical Einstein–Langevin equation. Due to the presence of the kernel , this equation will be usually non-local in the metric perturbation. In Section 6 we will carry out an explicit evaluation of the physical Einstein–Langevin equation which will illustrate the procedure.
When the expectation values in the Einstein–Langevin equation are taken in a vacuum state , such as, for instance, an “in” vacuum, we can be more explicit, since we can write the expectation values in terms of the Wightman and Feynman functions, defined as
From Equations (36), we see that the kernels and are the real and imaginary parts, respectively, of the bi-tensor . From the expression (4) we see that the stress-energy operator can be written as a sum of terms of the form , where and are some differential operators. It then follows that we can express the bi-tensor in terms of the Wightman function as
Similarly the kernel can be written in terms of the Feynman function as
Finally, the causality of the Einstein–Langevin equation (34) can be explicitly seen as follows. The non-local terms in that equation are due to the kernel which is defined in Equation (22) as the sum of and . Now, when the points and are spacelike separated, and commute and, thus, , which is real. Hence, from the above expressions, we have that , and thus . This fact is expected since, from the causality of the expectation value of the stress-energy operator , we know that the non-local dependence on the metric perturbation in the Einstein–Langevin equation, see Equation (14), must be causal. See  for an alternative proof of the causal nature of the Einstein–Langevin equation.
© Max Planck Society and the author(s)